(ii) In this case, we need to calculate the values of sinθ and cosθ first. They’ll come out to be 4/5 and 3/5 respectively. Now let’s use our formulas x = Xcosθ – Ysinθ and y = Xsinθ + Ycosθ.

That is, 3 = (3X – 4Y)/5 and 4 = (4X + 3Y)/5. Solving for X and Y, we get X = 5 and Y = 0. Therefore, the new coordinates will be (5, 0).

(iii) We didn’t talk about simultaneous rotation as well as translation. But turns out it is quite easy. We can find the new coordinates by first shifting the origin, followed by rotation, or the other way around.

We can also combine the two formulas straight away, i.e. x = Xcosθ – Ysinθ + h and y = Xsinθ + Ycosθ + k, and solve for X and Y to obtain the new coordinates. (You may try doing it separately and compare the answers)

Note that the axes are rotated clockwise in this case, but our formulas consider anticlockwise direction. So we’ve to take θ = –90°

This gives us X = – 6 and Y = 2. Therefore, the final coordinates are (–6, 2)

The next few problems will talk about equations of curves with respect to the new coordinate systems.

Example 2 Find the new equation of the following curves after the coordinates are transformed as indicated:

(i) x + 3y = 6, when the origin is shifted to the point (–4, 1).

(ii) Find the equation of the curve x2 + y2 = 4, when the axes are rotated by an angle of 60° in the anticlockwise direction.

(iii) Find the equation of the curve x2 – y2 = 10, when the axes are rotated by an angle of 45° in the clockwise direction.

Solution (i) In this case we do not need to find the new coordinates. We only need to find the relation between them (that’s what an equation is). So we’ll simply replace the old coordinates with the new ones in the given equation.

(iii) This one is quite similar to the previous one, except that we’re rotating the axes instead of translating them. Using the formulas, we have x = Xcos45° + Ysin45° and y = –Xsin45° + Ycos45°. On substituting in the given equation, we get (Xcos45° + Ysin45°)2 – (–Xsin45° + Ycos45°)2 = 10. This on simplification gives us XY = 5.

Example 3 To what point should the origin be shifted so that the equation x2 + y2 – 4x + 6y – 4 = 0 becomes free of the first degree terms? (i.e. -4x and 6y)

Example 4 By what angle should the axes be rotated so that the equation 3x2 + 2xy + y2 = 1 becomes free of the xy term?

Solution This one is similar to the previous one, except that now we’ve got to rotate the axes. Let’s do the hard work. The transformed equation will become 3(Xcosθ – Ysinθ)2 + 2(Xcosθ – Ysinθ)(Xsinθ + Ycosθ) + (Xsinθ + Ycosθ)2 = 1. (Whoa!)